686:
140:. This result was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his proof. Gelfand (independently) published a proof of the complex case a few years later.
1209:
1311:
575:
944:
411:
125:
The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals
238:
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401:
949:
722:
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528:
383:
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359:
1189:
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71:
1204:
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396:
224:
1341:
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815:
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190:. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:
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1027:
629:
151:
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8:
1089:
825:
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533:
247:
95:
1219:
1148:
1079:
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33:
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600:
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880:
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157:
60:
284:
1306:
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1199:
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1084:
1022:
1007:
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918:
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864:
789:
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467:
421:
369:
364:
335:
17:
294:
1256:
1235:
1153:
1143:
954:
861:
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309:
56:
41:
37:
29:
161:
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181:
94:
is not invertible. This is a consequence of the complex-analyticity of the
74:
of any element of a complex Banach algebra is nonempty: for every element
1271:
856:
580:
570:
477:
279:
764:
513:
345:
134:
52:
45:
749:
735:
1336:
1281:
49:
25:
246:
59:, i. e., the only complex Banach algebra that is a
1312:Spectral theory of ordinary differential equations
730:
576:Spectral theory of ordinary differential equations
1210:Schröder–Bernstein theorems for operator algebras
1354:
716:
232:
149:
70:The theorem follows from the fact that the
723:
709:
239:
225:
529:Group algebra of a locally compact group
150:Bonsall, Frank F.; Duncan, John (1973).
1355:
1043:Spectral theory of normal C*-algebras
841:Spectral theory of normal C*-algebras
704:
220:
180:
1038:Spectral theory of compact operators
192:McGraw-Hill Science/Engineering/Math
13:
1190:Cohen–Hewitt factorization theorem
114:1. This gives an isomorphism from
44:in which every nonzero element is
14:
1379:
1195:Extensions of symmetric operators
1013:Positive operator-valued measure
685:
684:
611:Topological quantum field theory
1368:Theorems in functional analysis
1297:Rayleigh–Faber–Krahn inequality
129:, the field of complex numbers
133:, and the division algebra of
1:
1205:Limiting absorption principle
407:Uniform boundedness principle
143:
82:there is some complex number
831:Singular value decomposition
78:of a complex Banach algebra
7:
1262:Hearing the shape of a drum
945:Decomposition of a spectrum
156:. Springer. pp. 71–4.
10:
1384:
850:Special Elements/Operators
550:Invariant subspace problem
1322:Superstrong approximation
1244:
1228:
1185:Banach algebra cohomology
1172:
1136:
1105:
1051:
1018:Projection-valued measure
1003:Borel functional calculus
995:
937:
894:
849:
803:
775:Projection-valued measure
742:
680:
639:
563:
542:
501:
440:
382:
328:
270:
263:
162:10.1007/978-3-642-65669-9
98:function. By assumption,
914:Spectrum of a C*-algebra
785:Spectrum of a C*-algebra
519:Spectrum of a C*-algebra
153:Complete Normed Algebras
1342:Wiener–Khinchin theorem
1277:Kuznetsov trace formula
1252:Almost Mathieu operator
1070:Banach function algebra
1059:Amenable Banach algebra
816:Gelfand–Naimark theorem
770:Noncommutative topology
616:Noncommutative geometry
63:is the complex numbers
1317:Sturm–Liouville theory
1215:Sherman–Takeda theorem
1095:Tomita–Takesaki theory
870:Hermitian/Self-adjoint
821:Gelfand representation
672:Tomita–Takesaki theory
647:Approximation property
591:Calculus of variations
811:Gelfand–Mazur theorem
667:Banach–Mazur distance
630:Generalized functions
22:Gelfand–Mazur theorem
1287:Proto-value function
1266:Dirichlet eigenvalue
1180:Abstract index group
1065:Approximate identity
1028:Rigged Hilbert space
904:Krein–Rutman theorem
750:Involution/*-algebra
412:Kakutani fixed-point
397:Riesz representation
102:1 −
90:1 −
36:which states that a
1090:Von Neumann algebra
826:Polar decomposition
596:Functional calculus
555:Mahler's conjecture
534:Von Neumann algebra
248:Functional analysis
187:Functional Analysis
40:with unit over the
1220:Unbounded operator
1149:Essential spectrum
1128:Schur–Horn theorem
1118:Bauer–Fike theorem
1113:Alon–Boppana bound
1106:Finite-Dimensional
1080:Nuclear C*-algebra
924:Spectral asymmetry
621:Riemann hypothesis
320:Topological vector
1350:
1349:
1327:Transfer operator
1302:Spectral geometry
987:Spectral abscissa
967:Approximate point
909:Normal eigenvalue
698:
697:
601:Integral operator
378:
377:
201:978-0-07-054236-5
171:978-3-642-65671-2
1375:
1332:Transform theory
1052:Special algebras
1033:Spectral theorem
996:Spectral Theorem
836:Spectral theorem
725:
718:
711:
702:
701:
688:
687:
606:Jones polynomial
524:Operator algebra
268:
267:
241:
234:
227:
218:
217:
213:
175:
61:division algebra
1383:
1382:
1378:
1377:
1376:
1374:
1373:
1372:
1363:Banach algebras
1353:
1352:
1351:
1346:
1307:Spectral method
1292:Ramanujan graph
1240:
1224:
1200:Fredholm theory
1168:
1163:Shilov boundary
1159:Structure space
1137:Generalizations
1132:
1123:Numerical range
1101:
1085:Uniform algebra
1047:
1023:Riesz projector
1008:Min-max theorem
991:
977:Direct integral
933:
919:Spectral radius
890:
845:
799:
790:Spectral radius
738:
732:Spectral theory
729:
699:
694:
676:
640:Advanced topics
635:
559:
538:
497:
463:Hilbert–Schmidt
436:
427:Gelfand–Naimark
374:
324:
259:
245:
202:
172:
146:
57:complex numbers
42:complex numbers
34:Stanisław Mazur
18:operator theory
12:
11:
5:
1381:
1371:
1370:
1365:
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1344:
1339:
1334:
1329:
1324:
1319:
1314:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1274:
1269:
1259:
1257:Corona theorem
1254:
1248:
1246:
1242:
1241:
1239:
1238:
1236:Wiener algebra
1232:
1230:
1226:
1225:
1223:
1222:
1217:
1212:
1207:
1202:
1197:
1192:
1187:
1182:
1176:
1174:
1170:
1169:
1167:
1166:
1156:
1154:Pseudospectrum
1151:
1146:
1144:Dirac spectrum
1140:
1138:
1134:
1133:
1131:
1130:
1125:
1120:
1115:
1109:
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1103:
1102:
1100:
1099:
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1097:
1087:
1082:
1077:
1072:
1067:
1061:
1055:
1053:
1049:
1048:
1046:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
999:
997:
993:
992:
990:
989:
984:
979:
974:
969:
964:
963:
962:
957:
952:
941:
939:
935:
934:
932:
931:
926:
921:
916:
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906:
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883:
875:
867:
859:
853:
851:
847:
846:
844:
843:
838:
833:
828:
823:
818:
813:
807:
805:
801:
800:
798:
797:
795:Operator space
792:
787:
782:
777:
772:
767:
762:
757:
755:Banach algebra
752:
746:
744:
743:Basic concepts
740:
739:
728:
727:
720:
713:
705:
696:
695:
693:
692:
681:
678:
677:
675:
674:
669:
664:
659:
657:Choquet theory
654:
649:
643:
641:
637:
636:
634:
633:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
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561:
560:
558:
557:
552:
546:
544:
540:
539:
537:
536:
531:
526:
521:
516:
511:
509:Banach algebra
505:
503:
499:
498:
496:
495:
490:
485:
480:
475:
470:
465:
460:
455:
450:
444:
442:
438:
437:
435:
434:
432:Banach–Alaoglu
429:
424:
419:
414:
409:
404:
399:
394:
388:
386:
380:
379:
376:
375:
373:
372:
367:
362:
360:Locally convex
357:
343:
338:
332:
330:
326:
325:
323:
322:
317:
312:
307:
302:
297:
292:
287:
282:
277:
271:
265:
261:
260:
244:
243:
236:
229:
221:
215:
214:
200:
177:
176:
170:
145:
142:
112:λ ·
38:Banach algebra
30:Israel Gelfand
9:
6:
4:
3:
2:
1380:
1369:
1366:
1364:
1361:
1360:
1358:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
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1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1295:
1293:
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1267:
1263:
1260:
1258:
1255:
1253:
1250:
1249:
1247:
1243:
1237:
1234:
1233:
1231:
1227:
1221:
1218:
1216:
1213:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1181:
1178:
1177:
1175:
1173:Miscellaneous
1171:
1164:
1160:
1157:
1155:
1152:
1150:
1147:
1145:
1142:
1141:
1139:
1135:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1110:
1108:
1104:
1096:
1093:
1092:
1091:
1088:
1086:
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1062:
1060:
1057:
1056:
1054:
1050:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
1000:
998:
994:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
961:
958:
956:
953:
951:
948:
947:
946:
943:
942:
940:
938:Decomposition
936:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
901:
899:
897:
893:
887:
884:
882:
879:
876:
874:
871:
868:
866:
863:
860:
858:
855:
854:
852:
848:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
808:
806:
802:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
747:
745:
741:
737:
733:
726:
721:
719:
714:
712:
707:
706:
703:
691:
683:
682:
679:
673:
670:
668:
665:
663:
662:Weak topology
660:
658:
655:
653:
650:
648:
645:
644:
642:
638:
631:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
586:Index theorem
584:
582:
579:
577:
574:
572:
569:
568:
566:
562:
556:
553:
551:
548:
547:
545:
543:Open problems
541:
535:
532:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
506:
504:
500:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
445:
443:
439:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
387:
385:
381:
371:
368:
366:
363:
361:
358:
355:
351:
347:
344:
342:
339:
337:
334:
333:
331:
327:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
272:
269:
266:
262:
257:
253:
249:
242:
237:
235:
230:
228:
223:
222:
219:
211:
207:
203:
197:
193:
189:
188:
183:
182:Rudin, Walter
179:
178:
173:
167:
163:
159:
155:
154:
148:
147:
141:
139:
136:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
68:
66:
62:
58:
54:
51:
50:isometrically
47:
43:
39:
35:
31:
27:
23:
19:
1245:Applications
1075:Disk algebra
929:Spectral gap
810:
804:Main results
652:Balanced set
626:Distribution
564:Applications
417:Krein–Milman
402:Closed graph
186:
152:
137:
130:
126:
124:
119:
115:
111:
107:
103:
99:
91:
87:
83:
79:
75:
69:
64:
28:named after
21:
15:
1272:Heat kernel
972:Compression
857:Isospectral
581:Heat kernel
571:Hardy space
478:Trace class
392:Hahn–Banach
354:Topological
135:quaternions
1357:Categories
950:Continuous
765:C*-algebra
760:B*-algebra
514:C*-algebra
329:Properties
144:References
86:such that
53:isomorphic
46:invertible
736:-algebras
488:Unbounded
483:Transpose
441:Operators
370:Separable
365:Reflexive
350:Algebraic
336:Barrelled
96:resolvent
1337:Weyl law
1282:Lax pair
1229:Examples
1063:With an
982:Discrete
960:Residual
896:Spectrum
881:operator
873:operator
865:operator
780:Spectrum
690:Category
502:Algebras
384:Theorems
341:Complete
310:Schwartz
256:glossary
210:21163277
184:(1991).
106:= 0. So
72:spectrum
878:Unitary
493:Unitary
473:Nuclear
458:Compact
453:Bounded
448:Adjoint
422:Min–max
315:Sobolev
300:Nuclear
290:Hilbert
285:Fréchet
250: (
55:to the
26:theorem
862:Normal
468:Normal
305:Orlicz
295:Hölder
275:Banach
264:Spaces
252:topics
208:
198:
168:
20:, the
955:Point
280:Besov
24:is a
886:Unit
734:and
628:(or
346:Dual
206:OCLC
196:ISBN
166:ISBN
32:and
158:doi
118:to
48:is
16:In
1359::
254:–
204:.
194:.
164:.
122:.
110:=
67:.
1268:)
1264:(
1165:)
1161:(
724:e
717:t
710:v
632:)
356:)
352:/
348:(
258:)
240:e
233:t
226:v
212:.
174:.
160::
138:H
131:C
127:R
120:C
116:A
108:a
104:a
100:λ
92:a
88:λ
84:λ
80:A
76:a
65:C
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